Lower Ricci curvature and nonexistence of manifold structure

Hupp, Erik; Naber, Aaron; Wang, Kai-Hsiang

doi:10.2140/gt.2025.29.443

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

Author Index

To Appear

Other MSP Journals

Lower Ricci curvature and nonexistence of manifold structure

Erik Hupp, Aaron Naber and Kai-Hsiang Wang

Geometry & Topology 29 (2025) 443–477

DOI: 10.2140/gt.2025.29.443

Abstract

We build limit spaces $(M_{j}^{n}, g_{j}) \to (X^{k}, d)$ of manifolds $M_{j}$ with uniform lower bounds on Ricci curvature such that $X^{k}$ is nowhere a topological manifold, and in fact every open set $U \subseteq X$ has infinitely generated homology.

More completely, it is known that any such $X^{k}$ must be $k$ -rectifiable for some unique $\dim X : = k \leq n = \dim M_{j}$ . It is also known that if $k = n$ , then $X^{n}$ is a topological manifold on an open dense subset, and it has been an open question as to whether this holds for $k < n$ . Consider now any smooth complete $4$ -manifold $(X^{4}, h)$ with $Ric > λ$ and $λ \in ℝ$ . Then for each $𝜖 > 0$ we construct a complete $4$ -rectifiable metric space $(X_{𝜖}^{4}, d_{𝜖})$ with $d_{G H} (X_{𝜖}^{4}, X^{4}) < 𝜖$ such that the following hold. First, $X_{𝜖}^{4}$ is a limit space $(M_{j}^{6}, g_{j}) \to X_{𝜖}^{4}$ , where $M_{j}^{6}$ are smooth manifolds satisfying the same lower Ricci bound ${Ric}_{j} > λ$ . Additionally, $X_{𝜖}^{4}$ has no open subset which is topologically a manifold. Indeed, for any open $U \subseteq X_{𝜖}^{4}$ we have that the second homology $H_{2} (U)$ is infinitely generated. Topologically, $X_{𝜖}^{4}$ is the connect sum of $X^{4}$ with an infinite number of densely spaced copies of $ℂ P^{2}$ .

In this way we see that every $4$ -manifold $X^{4}$ may be approximated arbitrarily closely by $4$ -dimensional limit spaces $X_{𝜖}^{4}$ which are nowhere manifolds. We will see that there is a sense, as yet imprecise, in which generically one should expect manifold structures to not exist on spaces with higher-dimensional Ricci curvature lower bounds.

Keywords

Ricci lower bounds, collapsing, non-manifold structure

Mathematical Subject Classification

Primary: 53B20, 53C21

Secondary: 51F30, 53C23, 58C07

References

Publication

Received: 3 September 2023

Revised: 2 February 2024

Accepted: 9 March 2024

Published: 22 January 2025

Proposed: Tobias H Colding

Seconded: Dmitri Burago, Bruce Kleiner

Authors

Erik Hupp
Department of Mathematics Northwestern University Evanston, IL United States

Aaron Naber
Institute for Advanced Study Princeton, NJ United States

Kai-Hsiang Wang
Department of Mathematics Northwestern University Evanston, IL United States

Open Access made possible by participating institutions via Subscribe to Open.

Volume 29, issue 1 (2025)